Problem: What is the sum of the integers that are both greater than 3 and less than 12?
Answer: We want to evaluate the arithmetic series $4+5+\dots+11$.

The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms.  The number of terms is $11 - 4 + 1 = 8$, so the sum is $(4 + 11)/2 \cdot 8 = \boxed{60}$.